LOWNESS FOR THE CLASS OF SCHNORR RANDOM
REALS
BJØRN KJOS-HANSSEN, ANDRÉ NIES, AND FRANK STEPHAN
Abstract. We answer a question of Ambos-Spies and Kučera in the
affirmative. They asked whether, when a real is low for Schnorr randomness, it is already low for Schnorr tests.
Keywords: lowness, randomness, Schnorr randomness, Turing degrees, recursion theory, computability theory. AMS subject classification 68Q30, 03D25, 03D28.
Contents
1. Introduction
2. Main concepts
2.1. Martingales
2.2. Traceability
3. Statement of the main result
4. Proof of the main result
5. Lowness notions related to Chaitin’s halting probability
References
1
4
4
4
5
6
12
13
1. Introduction
In an influential 1966 paper [9], Martin-Löf proposed an algorithmic formalization of the intuitive notion of randomness for infinite sequences of 0’s
and 1’s. His formalization was based on an effectivization of a test concept from statistics, by means of uniformly recursively enumerable (r.e.) sequences of open sets. Martin-Löf’s proposal addressed some insufficiencies
in an earlier algorithmic concept of randomness proposed by Church [3],
who had formalized a notion now called computable stochasticity. However, Schnorr [13] criticized Martin-Löf’s notion as too strong, because it
was based on an r.e. test concept rather than a computable notion of tests.
Department of Mathematics, University of Connecticut, Storrs, CT 06269
([email protected]. edu).
Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand ([email protected]).
School of Computing and Department of Mathematics, National University of Singapore, 3 Science Drive 2, Singapore 117543, Republic of Singapore ([email protected].
edu.sg).
1
2
B. KJOS-HANSSEN, A. NIES, AND F. STEPHAN
He suggested that one should base a formalization of randomness on computable betting strategies (also called martingales), in a way that would still
overcome the problem that Church’s concept was too weak. In present terminology, a real Z is computably random if no computable betting strategy
succeeds along Z; that is, for each computable betting strategy there is a
finite upper bound on the capital that it reaches. The real Z is Schnorr
random if no martingale succeeds effectively. Here effective success means
that the capital at Z n exceeds f (n) infinitely often, for some unbounded
computable function f . See [1] for more on the history of these ideas.
We recall some definitions. The Cantor space 2ω is the set of infinite
binary sequences; these are called reals and are identified with a set of
integers, i.e., subsets of ω. If σ ∈ 2<ω , that is, σ is a finite binary sequence,
then we denote by [σ] the set of reals that extend σ. These form a basis of
clopen sets for the usual discrete topology on 2ω . Write |σ| for the length
of σ ∈ 2<ω . The Lebesgue measure µ on 2ω is defined by stipulating that
µ[σ] = 2−|σ| . With every set U ⊆ 2<ω we associate the open set [U ] =
S
<ω and σ is a prefix of
σ∈U [σ]. The empty sequence is denoted λ. If σ, τ ∈ 2
<ω
τ , then we write σ τ . If σ ∈ 2 and i ∈ {0, 1}, then σi denotes the string
of length |σ|+1 extending σ whose final entry is i. The concatenation of two
strings σ and τ is denoted στ . The empty set is denoted ∅, and inclusion of
sets is denoted by ⊆. If A is a real and n ∈ ω, then A n is the prefix of A
consisting of the first n bits of A. Letting A(n) denote bit n of A, we have
A n = A(0)A(1) · · · A(n − 1).
Given α ∈ 2<ω and a measurable set C ⊆ 2ω , we let µα C = µ(C∩[α])
µ[α] . For
an open set W we let
W |σ =
[
[τ ] : τ ∈ 2<ω , [στ ] ⊆ W .
Note in particular that µσ W = µ(W |σ) and µλ W = µW .
Fixing some effective correspondence between the set of finite subsets of ω
and ω, we let De be the eth finite subset of ω under this correspondence. In
other words, e is a strong, or canonical, index for the finite set De . Similarly,
we let Se be the eth finite subset of 2<ω under a suitable correspondence.
Thus Se is a finite set of strings, and [Se ] = ∪σ∈Se [σ] is then the clopen set
coded by e ∈ ω. We use the Cantor pairing function, namely the bijection
2
p : ω 2 → ω given by p(n, s) = (n+s) 2+3n+s , and write hn, si = p(n, s).
A Martin-Löf test is a set U ⊆ ω × 2ω such that µUn ≤ 2−n , where Un
denotes the nth section of U , and Un is a Σ01 class, uniformly in n. If,
in addition, µUn is a computable real, uniformly in n, then U is called a
Schnorr test. Z is Martin-Löf random if for each Martin-Löf test U there
is an n such that Z 6∈ Un , and Schnorr random if for each Schnorr test
U there is an n such that Z 6∈ Un . The notion of Schnorr randomness is
unchanged if we instead define a Schnorr test to be a Martin-Löf test for
which µUn = 2−n for each n ∈ ω.
SCHNORR RANDOM REALS
3
Concepts encountered in computability theory are usually based on some
notion of computation, and therefore have relativized forms. For instance,
we may relativize the tests and randomness notions above to an oracle A. If
C = {X : X is Martin-Löf random}, then the relativization is CA = {X : X
is Martin-Löf random relative to A} (meaning that Σ01 classes are replaced
by Σ0,A
classes). In general, if C is such a relativizable class, we say that
1
A is low for C if CA = C. If C is a randomness notion, more computational
power means a smaller class, namely CA ⊆ C for any A. Being low for
C means having small computational power (in a sense that depends on
C). In particular, the low-for-C reals are closed downward under Turing
reducibility.
The randomness notions for which lowness was first considered are MartinLöf and Schnorr randomness. Kučera and Terwijn [8] constructed a noncomputable r.e. set of integers A which is low for Martin-Löf randomness,
answering a question of Zambella [16]. In the paper [14] it is shown that
there are continuously many reals that are low for Schnorr randomness.
An important difference between the two randomness notions is that for
Martin-Löf randomness, but not for Schnorr randomness,
T there is a universal
test R. Thus, Z is not Martin-Löf random iff Z ∈ b∈ω Rb . Therefore, in
the Schnorr case, an apparently stronger lowness notion is being low for
Schnorr tests, or S0 -low in the terminology of [1]: A is low for Schnorr tests
if for each SchnorrTtest U A T
relative to A there is an unrelativized Schnorr
test V such that n UnA ⊆ n Vn . This implies that A is low for Schnorr
randomness, or S-low in the terminology of [1]. Ambos-Spies and Kučera
asked if the two notions coincide. We answer this question in the affirmative.
Terwijn and Zambella [14] actually constructed oracles A that are low for
Schnorr tests. They first gave a characterization of this lowness property
via a notion of traceability, a restriction on the possible sequence of values
of the functions computable from A. They showed that A is low for Schnorr
tests iff A is computably traceable (see formal definition in the next section).
Then they constructed continuously many computably traceable reals. We
answer the question of Ambos-Spies and Kučera by showing that each real
which is low for Schnorr randomness is in fact computably traceable.
Towards this end, it turns out to be helpful to have a more general view
of lowness. We consider lowness for any pair of randomness notions C, D
with C ⊆ D.
Definition 1.1. A is in Low(C, D) if C ⊆ DA . We write Low(C) for
Low(C, C).
e ⊆D
e ⊆ D are randomness notions, and the inclusions
Clearly, if C ⊆ C
A
e
e D)
e ⊆ Low(C, D).
relativize (so D ⊆ DA for each real A), then Low(C,
e D)
e larger by decreasing C or increasing
That is, we make the class Low(C,
D. Let MR, CR, and SR denote the classes of Martin-Löf random, computably random (defined below), and Schnorr random reals, respectively. Thus,
4
B. KJOS-HANSSEN, A. NIES, AND F. STEPHAN
for instance, Low(MR, CR) is the class of oracles A such that each MartinLöf random real is computably random in A. We will characterize lowness
for any pair of randomness notions C ⊆ D with C, D ∈ {MR, CR, SR}.
Recall that Ω denotes the halting probability of a universal prefix machine.
Ω is a Martin-Löf random r.e. real, i.e., a real that can be effectively approximated from below. Given D ⊇ MR, an interesting lowness notion obtained
by weakening Low(MR, D) is Low({Ω}, D). That is, instead of MR ⊆ DA
one merely requires that Ω ∈ DA . We denote this class by Low(Ω, D). In
[12], the case D = MR is studied. The authors show that the class coincides
with Low(MR) on the ∆02 reals but not in general. In fact, a Martin-Löf
random real is 2-random iff it is in Low(Ω, MR).
Here we investigate the class Low(Ω, SR). We show that A is Low(MR,
SR) iff A is r.e. traceable. Moreover, the weaker assumption Ω ∈ SRA still
implies that A is array computable (there is a function f ≤wtt ∅0 bounding
all functions computable from A, on almost all inputs). Thus for r.e. sets
of integers A, A being Low(MR, SR) is in fact equivalent to Ω ∈ SRA by
Ishmukhametov [5]. We also provide an example of a real A which is array
computable but not Low(Ω, SR).
2. Main concepts
2.1. Martingales. For our purposes, a martingale is a function M : 2<ω 7→
Q (where Q is the set of rational numbers) such that (i) the domain of M
is 2<ω , or 2≤n = {σ ∈ 2<ω : |σ| ≤ n} for some n, (ii) M (λ) ≤ 1, and (iii) M
has the martingale property M (x0) + M (x1) = 2M (x) whenever the strings
x0, x1 belong to the domain of M . A martingale M succeeds on a sequence
Z ∈ 2ω if
lim sup M (Z n) = ∞.
n→∞
A real is computably random if no computable martingale succeeds on it.
A martingale M effectively succeeds on a sequence Z if there is a nondecreasing and unbounded computable function h : ω −→ ω such that
lim sup M (Z n) − h(n) > 0.
n→∞
Equivalently (since we are considering integer-valued functions), ∃∞ n M (Z n) > h(n). We can now state the characterization of Schnorr randomness
in terms of martingales: a real Z is Schnorr random iff no computable
martingale effectively succeeds on Z.
2.2. Traceability. Let We denote the eth r.e. set of integers in some standard list. A real A is r.e. traceable if there is a computable function p, called
a bound, such that for every f ≤T A there is a computable function r such
that for all x we have |Wr(x) | ≤ p(x) and f (x) ∈ Wr(x) .
The following is a stronger notion than r.e. traceability. A is computably
traceable if there is a computable p such that for every f ≤T A there is a
computable r such that for all x we have |Dr(x) | ≤ p(x) and f (x) ∈ Dr(x) .
SCHNORR RANDOM REALS
5
It is interesting to notice that it does not matter what bound p one chooses
as a witness for traceability; see the following.
Proposition 2.1 (see Terwijn and Zambella [14]). Let A be a real that is
computably traceable with bound p. Then for any monotone and unbounded
computable function p0 , A is computably traceable with bound p0 . The same
holds for r.e. traceability.
The result of Terwijn and Zambella is the following.
Theorem 2.2 (see [14]). A real A is low for Schnorr tests iff A is computably traceable.
3. Statement of the main result
Theorem 3.1.
(I) A is Low(MR, SR) iff A is r.e. traceable.
(II) A is Low(CR, SR) iff A is Low(SR) iff A is computably traceable.
We make some remarks about the proofs and fill in the details in the next
section. We obtain Theorem 3.1(I) by modifying the methods in [14] to the
case of r.e. traces instead of computable ones.
As for Theorem 3.1(II), by Theorem 2.2 if A is computably traceable, then
A is low for Schnorr tests. Hence A is certainly Low(SR), and therefore also
Low(CR, SR). It remains only to show that each real A ∈ Low(CR, SR) is
computably traceable. To see that this is so, take the following three steps:
1. Recall that A is hyperimmune-free if for each g ≤T A there is a
computable f such that for all x we have g(x) ≤ f (x). As a first step
towards proving Theorem 3.1(II), Bedregal and Nies [2] showed that each
A ∈ Low(CR, SR) is hyperimmune-free (see Lemma 4.9 below). To see this,
assume that A is not, so there is a function g ≤T A not dominated by any
computable function f . Define a martingale L ≤T A which succeeds in the
sense of Schnorr, with the computable lower bound h(n) = n/4, on some
Z ∈ CR. One uses here that g is infinitely often above the running time of
each computable martingale. (Special care has to be taken with the partial
martingales, which results in a real Z that is only ∆03 .)
2. If A is hyperimmune-free and r.e. traceable, then A is computably
traceable. If we let g ≤T A, then the first stage where g(x) appears in a
given trace for g can be computed relative to A.
3. Now each A in Low(CR, SR) is r.e. traceable by Theorem 3.1(I), and
hence by the above is computably traceable, and Theorem 3.1(II) follows.
We discuss lowness for the remaining pairs of randomness notions. Nies
has shown that A is Low(MR, CR) iff A is Low(MR) iff A is K-trivial, where
A is K-trivial if for all n K(X n) ≤ K(n) + O(1) (see [11]). Here K(σ)
denotes the prefix-free Kolmogorov complexity of σ ∈ 2<ω . Finally, he
shows that a real A which is Low(CR) is computable; namely, A is both
6
B. KJOS-HANSSEN, A. NIES, AND F. STEPHAN
K-trivial and hyperimmune-free. Since all K-trivial reals are ∆02 , and all
hyperimmune-free ∆02 reals are computable, the conclusion follows.
4. Proof of the main result
We first need to develop a few useful facts from measure theory.
Definition 4.1. A measurable set A has density d at a real X if
lim µ(Xn) A = d.
n→∞
A basic result is the following.
Theorem 4.2 (Lebesgue density theorem). Let Ξ(A) = {X : A has density
1 at X}. If A is a measurable set, then so is Ξ(A), and the measure of the
symmetric difference of A and Ξ(A) is zero.
Corollary 4.3. Let C be a measurable subset of 2ω , with µC > 0. Then for
each δ < 1 there is an α ∈ 2<ω such that µα C ≥ δ.
We will use the following consequence of Corollary 4.3.
Lemma T4.4. Let 0 < ≤ 1. If Un , n ∈ ω, and V are open subsets of
2ω with n∈ω Un ⊆ V and µV < , then there exist σ and n such that
µσ (Un − V ) = 0 and µσ V < .
Proof. Suppose
otherwise; we shall obtain a contradiction by constructing a
T
real in n∈ω Un − V . Let σ0 = λ and assume we have defined σn such that
µσn V < . By hypothesis, µσn (Un −V ) > 0, and thus there is a [τ ] ⊆ Un such
that µσn ([τ ] − V ) > 0. In particular, τ σn and µτ V < 1. Let C = 2ω − V ,
a closed and hence measurable set. By Corollary 4.3 applied to C (and with
2ω replaced by [τ ]), there exists σn+1 τ such that µσn+1 V < . Let X be
the real that extends all σ
Tn ’s constructed in this way. Since [σn+1 ] ⊆ Un for
all n, we have that X ∈ n∈ω Un . However, [σn ] 6⊆ V for every n, so, since
V is open, X 6∈ V . This contradiction completes the proof.
We now get to the proof of Theorem 3.1. First we show Theorem 3.1(I),
namely, that A is Low(MR, SR) iff A is r.e. traceable. We start with the
“⇐” direction.
Lemma 4.5. If A is r.e. traceable, then A is Low(MR, SR).
Proof. Assume that A is r.e. traceable and that U A is a Schnorr test
S relative
A , n, s ∈ ω, be clopen sets, U A ⊆ U A
A =
A
to A. Let Un,s
,
U
n,s
n
n,s+1
s∈ω Un,s ,
A are ∆0,A classes uniformly in n and s. As µU A = 2−n , we
such that the Un,s
n
1
A > 2−n (1 − 2−s ). Let f be an A-computable function
may assume that µUn,s
A . Since A is r.e. traceable and f ≤ A, we can
such that [Sf (hn,si) ] = Un,s
T
let T be an r.e. trace of f . By Proposition 2.1, we may choose T such that
in addition |Tx | ≤ x for each x > 0.
SCHNORR RANDOM REALS
7
We now want to define a subtrace T̂ of T , i.e., T̂hn,si ⊆ Thn,si for each
n, s. The intent is that the open sets defined via T̂ are small enough to
give us a Martin-Löf test containing ∩n∈ω UnA , and nothing important is in
Thn,si − T̂hn,si . Thus let T̂hn,si be the set of e ∈ Thn,si such that 2−n (1−2−s ) ≤
µ[Se ] ≤ 2−n and [Se ] ⊇ [Sd ] for some d ∈ T̂hn,s−1i , where T̂hn,−1i = ω.
Let
o
[n
Vn =
[Se ] : e ∈ T̂hn,si , s ∈ ω .
P
Then µVn ≤ 2−n |T̂hn,0i | + s∈ω 2−s 2−n |T̂hn,si |. Since |T̂hn,si | ≤ |Thn,si | ≤
hn, si for hn, si 6= 0, and hn, si has only polynomial growth in n and s, it is
P
clear that s∈ω 2−s 2−n |T̂hn,si | is finite and goes effectively to 0 as n → ∞;
hence the same can be said of µVn . Thus there is a recursive function f
such that µVf (n) ≤ 2−n . Let Ṽn = Vf (n) . Then Ṽ is a Martin-Löf test and
T
T A
n Ṽn . That is, each Schnorr test relative to A is contained in
n Un ⊆
a Martin-Löf test. It follows that each real that is Martin-Löf random is
Schnorr random relative to A, and the proof is complete.
Next we will show the “⇒” direction of Theorem 3.1(I). The proof is
similar to the “⇒” of Theorem 2.2.
Definition 4.6. For k, l ∈ ω define the clopen set
o
[n
Bk,l =
[τ 1k ] : τ ∈ 2<ω , |τ | = l ,
where 1k is a string of 1’s of length k.
Note that µBk,l = 2−k for all l.
Lemma 4.7. If A ∈ 2ω is Low(MR, SR), then A is r.e. traceable.
A
Proof.
T A is Low(MR, SR) iff for every Schnorr test U relative to
T NoteA that
A, n∈ω Un ⊆ b∈ω Rb (recall that R is a universal Martin-Löf test).
Oversimplifying a bit, one can say that the proof below goes as follows.
g
We code a given
T A into a Schnorr test U relative to A. Then,
T g by
T gg ≤T
hypothesis, n Un ⊆ n Rn ; in fact we will use only the fact that n Un ⊆
R3 . We then define an r.e.trace T ; namely, Tk is the set of l such that
Bk,l − R3 has small measure in some sense. Since R3 has rather small
measure, Bk,l − R3 will tend to have big measure, which means that there
will be only a few l for which Bk,l − R3 has small measure; in other words,
Tk has small size. Moreover, we make sure T is a trace for g so that A is
r.e. traceable.
We now give the proof details. Suppose that we want to find a trace for
a given function g ≤T A. We define the test U g by stipulating that
[
Ung =
Bk,g(k) .
k>n
µUng
It is easy to see that
can be approximated computably in A, so after
taking a subsequence of Ung , n ∈ ω, we may assume that U g is a Schnorr
8
B. KJOS-HANSSEN, A. NIES, AND F. STEPHAN
T
T
test relative to A. Hence, by assumption, U g ⊆ b∈ω Rb . Thus V = R3
T g
contains U and µV < 14 . We may assume throughout that g(k) ≥ k
for every k because from a trace for g(k) + k one can obtain a trace for
g with the same bound. By Lemma 4.4, there exist σ and n such that
µσ (Ung − V ) = 0 and µσ V < 1/4. As U0g ⊇ U1g ⊇ · · · , we can choose σ
and n with the additional property n ≥ |σ|. Hence for each k > n, we have
g(k) ≥ k > n ≥ |σ| and hence g(k) ≥ |σ|.
Let Ṽ = V |σ, let g̃(k) = max{0, g(k) − |σ|}, and take
n
o
Tk = l : µ(Bk,l − Ṽ ) < 2−(l+3) .
Note that for each l ∈ ω, if l ≥ |σ|, then Bk,l |σ = Bk,l−|σ| . Thus, since
g(k) ≥ |σ|,
[
[
Ung |σ =
Bk,g(k) |σ =
Bk,g(k)−|σ| = Ung̃ ,
k>n
µσ (Ung
µ(Ung̃
k>n
and so
− Ṽ ) =
− V ) = 0. Hence g̃(k) ∈ Tk for all k > n.
Since Ṽ is a Σ01 class, it is evident that T is an r.e. set of integers; indeed
Bk,l − Ṽ is a Π01 class, and thus we can enumerate the fact that certain basic
open sets [σ] are disjoint from it, until the measure remaining is as small as
required. A trace for g is obtained as follows:
(
{l + |σ| : l ∈ Tk } if k > n,
Gk =
{g(k)}
if k ≤ n.
We now show that G is a trace for g; i.e., for all k ∈ ω, g(k) ∈ Gk .
If k ≤ n, then this holds by definition of Gk ; thus suppose k > n. Then
g(k) > k > n > |σ|, so g̃(k) = g(k) − |σ|, so g(k) = g̃(k) + |σ|. As k > n,
g̃(k) ∈ Tk and hence g(k) ∈ Gk .
Clearly G is r.e.; thus it remains to show that |Gk | is computably bounded,
independently of g. As |Gk | = |Tk | for k > n and |Gk | = 1 for k ≤ n, this is
a consequence of Lemma 4.8 below.
Lemma 4.8. If Ṽ is a measurable set with µṼ < 41 , and Tk = {l : µ(Bk,l −
Ṽ ) < 2−(l+3) }, then for k ≥ 1, |Tk | < 2k k.
Proof. Observe that, by definition of Tk ,
X
X
1 X −l 1
µ(Bk,l − Ṽ ) <
2−(l+3) ≤
2 = ,
8
4
l∈Tk
l∈ω
l∈Tk
so
µ
[
Bk,l − µṼ ≤ µ
l∈Tk
As µṼ <
1
4,
[
l∈Tk
1
(Bk,l − Ṽ ) ≤ .
4
we obtain that
µ
[
l∈Tk
1
Bk,l < .
2
SCHNORR RANDOM REALS
9
As observed above, µBk,l = 2−k . Moreover, for k fixed, the Bk,l ’s are
mutually independent as soon as the l’s are taken sufficiently far apart. In
fact, sufficiently far here means a distance of k. So for k ≥ 1 we let Tk∗
be a subset of Tk consisting of |Tkk | elements, all of which are sufficiently
far apart. (Here bac is the greatest integer ≤ a.) To show that such a set
exists we may assume we are in the worst case, where the elements of Tk are
closest together: say, Tk = {0, . . . , |Tk | − 1}. Then let Tk∗ = {mk : 0 ≤ m ≤
|Tk | − 1}. As |Tkk | − 1 k ≤ |Tk | − k ≤ |Tk | − 1 ∈ Tk , this makes Tk∗ ⊆ Tk .
k
Write α = |Tkk | . We now have
\
\
(2ω − Bk,l ) ≤ µ
(2ω − Bk,l ) = (1 − 2−k )α
µ
l∈Tk∗
l∈Tk
and hence
\
1 − (1 − 2−k )α ≤ 1 − µ
(2ω − Bk,l ) = µ2ω − µ
≤ µ 2ω −
(2ω − Bk,l )
l∈Tk
l∈Tk

\

\
l∈Tk
(2ω − Bk,l ) = µ
[
l∈Tk
1
Bk,l < .
2
From the inequality above we obtain, letting m = 2k − 1,
α
1
1
= (1 − 2−k )α >
1−
m+1
2
α
m
α
or m+1
< 2. Now suppose α ≥ m. Then m+1
≥ m+1
≥ 2 as
m
m
m
m . Thus we conclude α < m = 2k − 1.
=
2m
(m + 1)m ≥ mm + mm−1 m
1
Now, by the definition of α, we have Tkk ≤ α + 1 < 2k and so |Tk | < 2k k;
this completes the proof.
In order to prove Theorem 3.1(II), recall that, by Theorem 2.2, each
computably traceable real is Low(SR). Thus it suffices to show that each
Low(CR, SR) real is computably traceable. The first ingredient for showing
this is the following result from [2].
Lemma 4.9. If A is Low(CR, SR), then A is hyperimmune-free.
Proof. Suppose A is not hyperimmune-free, so that there is a function
g ≤T A not dominated by any computable function. Thus, for each computable f , ∃∞ x f (x) ≤ g(x). We will define a computably random real
X and an A-computable martingale L that succeeds on X in the sense of
Schnorr, so that A is notPLow(CR, SR). In the following, α, β, γ denote finite
subsets of ω, and nα = i∈α 2i (here n∅ = 0).
Let {Me }e∈ω be an effective listing of all partial computable martingales
with range included in [1/2, ∞). At stage t, we have a finite portion Me [t],
whose domain is a subset of some set of the form 2≤n for some n. If X is
10
B. KJOS-HANSSEN, A. NIES, AND F. STEPHAN
not computably random, then limn→∞ Me (X n) = ∞ for some total Me
by [13]. Let
T M G = {e : Me is total}.
For finite sets α, β, let us in this proof say that α is a strong subset of β
(denoted α ⊆+ β) if α ⊆ β and moreover for each i ∈ ω, if i ∈ β − α, then
i > max(α). Thus the possibility that β contains an element smaller than
some element of α is ruled out.
For certain α, and all those included in T M G, we will define strings xα
in such a way that α ⊆+ β ⇒ xα xβ . We choose the strings in such a way
that Me (xα ) is bounded by a fixed constant (depending on e) for each total
Me and each α containing e. Then the set of integers
[
X=
xT M G∩[0,e]
e∈ω
is a computably random real. On the other hand, we are able to define
an A-computable martingale L which Schnorr succeeds on X. We give an
inductive definition of the strings xα , “scaling factors” pα ∈ Q+ (positive
rationals) (we do not define p∅ ), and partial computable martingales Mα
such that if xα is defined, then
(1)
Mα (xα ) converges in g(|xα |) steps and Mα (xα ) < 2.
It will be clear that A can decide if y = xα , given inputs y and α.
Let x∅ = λ, and let M∅ be the constant zero function. (We may assume
that g is such that M∅ (λ) converges in g(0) steps.) Now suppose α = β ∪{e},
where e > max(β), and inductively suppose that (1) holds for β. Let
1
pα = 2−|xβ | (2 − Mβ (xβ )),
2
and let Mα = Mβ + pα Me . Since Me is a martingale on its domain, Me (z) ≤
2|z| for any z. So, writing b = Mβ (xβ ), we have Mα (xβ ) = b + pα Me (xβ ) <
b + pα 2|xβ | = b + 21 (2 − b) = 1 + 2b < 1 + 22 = 2 if Mα (xβ ) is defined.
To define xα , we look for a sufficiently long x xβ such that Mα does
not increase from xβ to x and Mα (x) converges in g(|x|) steps. In detail,
for larger and larger m > |xβ |, m ≥ 4nα , if no string y, |y| < m, has been
designated to be xα as yet, and if Mα (z) (i.e., each Me (z), e ∈ α) converges
in g(m) steps, for each string z of length ≤ m, then choose xα of length m,
xβ ≺ xα such that Mα does not increase anywhere from xβ to xα .
Claim 4.10. If α ⊆ T M G, then xα and pα (the latter only if α 6= ∅) are
defined.
Proof. The claim is trivial for α = ∅. Suppose that it holds for β, and
α = β ∪ {e} ⊆ T M G, where e > max(β). Since the function
f (m) = µs
∀e ∈ α, ∀x
[ |x| ≤ m ⇒ Me (x) converges in s steps ]
SCHNORR RANDOM REALS
11
is computable, there is a least m ≥ 4nα , m > |xβ | such that g(m) ≥ f (m).
Since there is a path down the tree starting at xβ , where Mα does not
increase, the choice of xα can be made.
Claim 4.11. If β ⊆+ α are finite sets, then Mβ (x) ≤ Mα (x) for all x.
Proof. This is clear by induction from the case α = β ∪ {e}, i.e., the case
where α − β has only one element.
Claim 4.12. X is computably random.
Proof. Suppose that Me is total. Let α = T M G ∩ [0, e]. Suppose α ⊆ γ,
γ 0 = γ ∪ {i}, max(γ) < i, and γ 0 ⊆ T M G. Then α ⊆+ γ ⊆+ γ 0 . Hence by
Claim 4.11, for each x with xγ x xγ 0 , we have
pα Me (x) ≤ Mα (x) ≤ Mγ 0 (x) ≤ Mγ 0 (xγ ) < 2,
and hence Me (x) < 2/pα for each x ≺ X, and so the capital of Me on X is
bounded.
Claim 4.13. There is a martingale L ≤T A which effectively succeeds on
X. In fact,
∞
∃ x ≺ X L(x) ≥
|x|
.
4
P
Proof. For a string z, let r(z) = b|z|/2c. We let L =
α Lα , where Lα
is a martingale with initial capital Lα (λ) = 2−nα , which bets everything
along xα from xα r(xα ) on. More precisely, if xα is undefined, then Lα is
constant with value 2−nα . Otherwise, for convenience we let x = xα 2r(xα )
and work with x instead of xα ; we define Lα on a string y as follows:
• If y does not contain “half of x,” i.e., if x r(x) 6 y, then just let
Lα (y) = 2−nα .
• If y does contain “half of x” but y and x are incompatible, then let
Lα (y) = 0.
• If y contains “half of x” and x and y are compatible, then let Lα (y) =
2−nα 2min(|y|−r(x),r(x)) .
Thus if y contains x, then Lα (y) = 2r(x)−nα , so we make no more bets
once we extend xα , and if x contains y, then Lα (y) = 2|y|−r(x)−nα ; i.e., we
double the capital for P
each correct bit of x beyond x r(x).
−nα , and, as each k ∈ ω has a unique binary
Note that L(λ) =
α2
expansion
and hence is equal to nα for a unique finite set α, we have L(λ) =
P
−k = 2. Moreover, it is clear that each L satisfies the martingale
2
α
k∈ω
property Lα (x0) + Lα (x1) = 2Lα (x), and hence so does L.
L effectively succeeds on X. Indeed, as |xα | ≥ 4nα , we have Lα (xα ) =
r(x
2 α )−nα ≥ 2b|xα |/2c−b|xα |/4|c ≥ 2b|xα |/4c ≥ b|xα |/4c since 2q ≥ q for each
q ∈ ω.
12
B. KJOS-HANSSEN, A. NIES, AND F. STEPHAN
Finally, we show that L ≤T A. Given input y, we use g to see if some
string x, |x| ≤ 2|y|, is xα . If not, Lα (y) = 2−nα . Else we determine Lα (y)
from x using the definition of Lα .
The second ingredient to the proof of Theorem 3.1(II) is the following
fact of independent interest.
Proposition 4.14. If A is hyperimmune-free and r.e. traceable, then A is
computably traceable.
Proof. Let f ≤T A, and let h be as in the definition of r.e. traceability.
Let g(x) = µs(f (x) ∈ Wh(x),s ) (where We,s is the approximation at stage
s to the r.e. set We ). Then g ≤T A, and so since A is hyperimmune-free,
g is dominated by a computable function r. Thus if we replace Wh(x) by
Wh(x),r(x) , we obtain a computable trace for f .
Lemma 4.9 and Proposition 4.14 together establish Theorem 3.1(II): if A is
Low(CR, SR), then A is r.e. traceable by Theorem 3.1(I), and hyperimmunefree by Lemma 4.9. Thus by Proposition 4.14, A is computably traceable.
As a corollary, we obtain an answer to the question of Ambos-Spies and
Kučera.
Corollary 4.15. A real A is S-low iff it is S0 -low.
Proof. This follows by Theorem 2.2 and Theorem 3.1(II), since each computably traceable real is S0 -low.
5. Lowness notions related to Chaitin’s halting probability
Recall that A is array computable if there is a function f ≤wtt ∅0 bounding
all functions computable from A on almost all inputs.
Theorem 5.1. If Ω ∈ SRA , then A is array computable.
Proof. We show that the function β(x) = µs Ωs 3x = Ω 3x dominates
each function α ≤T A. Since β ≤wtt Ω ≤wtt 00 , this shows that A is array
computable.
P
Given α ≤T A, consider the A-computable martingale M =
p Mp ,
−p
where Mp is the martingale which has the value 2 on all strings of length
up to p and then doubles the capital along the string y = Ωα(p) 3p, so
that Mp (y) = 2p . Note that M (z) is rational for each z. If α(p) > β(p) for
infinitely many p, then M Schnorr succeeds on Ω, a contradiction.
Corollary 5.2. If A is r.e., then Ω ∈ SRA iff A is r.e. traceable.
Proof. For an r.e. set A, array computable implies r.e. traceable by the work
of Ishmukhametov [5].
In [6] it is shown that r.e. traceable degrees do not contain diagonally
noncomputable functions, and hence, by a result of Kučera [7], the r.e.
traceable degrees have measure zero. On the other hand, every real A which
SCHNORR RANDOM REALS
13
is Martin-Löf random relative to Ω satisfies that Ω is MRA , by van Lambalgen’s theorem [15], and hence the measure of the set of A such that Ω is SRA
is one; thus A r.e. traceable is not equivalent to Ω ∈SRA . Also, Ω ∈SRA is
not equivalent to A being array computable, as we now show.
The following notion of forcing appears implicitly in [4].
Definition 5.3. A tree T is a set of strings σ ∈ 2<ω such that if σ ∈ T and τ
is a substring of σ, then τ ∈ T . A tree T is full on a set F ⊆ ω if whenever
σ ∈ T and |σ| ∈ F , then σ0 ∈ T and σ1 ∈ T . Let Fn , n ∈ S
ω, be finite
sets such that each Fn is an interval of ω, |Fn+1 | > |Fn |, and n Fn = ω.
The sequence Fn , n ∈ ω, is called a very strong array. Let P be the set of
computable perfect trees T such that T is full on Fn for infinitely many n.
Order P by T1 ≤P T2 if T1 ⊆ T2 . The partial order (P, ≤P ) is a notion of
forcing that we call very strong array forcing.
Theorem 5.4. For each real X there is a hyperimmune-free real A such
that no real computable from X is in SRA . In particular, as hyperimmunefree implies array computable, there is an array computable real A such that
Ω 6∈ SRA .
Proof. Let A be sufficiently generic for very strong array forcing. Then A is
hyperimmune-free, as may be proved by modifying the standard construction of a hyperimmune-free degree [10] to work with trees that are full on
infinitely many Fn , n ∈ ω.
Moreover, for each real B computable from X, there is an n (hence infinitely many n) such that A agrees with B on Fn . Indeed, given a condition T ,
a condition extending T and ensuring the existence of such an n is obtained
as a full subtree of T .
Hence no real B computable from X is Schnorr random relative to A.
Indeed the measure of the set of those oracles B that agree with A on
infinitely many Fn is zero, and it is easy to see that the measure of those B
such that, for some k > n, A and B agree on Fk , goes to zero effectively as
n → ∞. Hence there is a Schnorr test relative to A which is failed by any
such B, as desired.
Question 5.5. Characterize the (r.e.) sets of integers A such that Ω is
computably random relative to A. Does this depend on the version of Ω
used?
References
[1] K. Ambos-Spies and A. Kučera, Randomness in computability theory, in Computability Theory and Its Applications (Boulder, CO, 1999), Contemp. Math. 257,
AMS, Providence, RI, 2000, pp. 1–14.
[2] B. Bedregal and A. Nies, Lowness properties of reals and hyper-immunity, in Proceedings of the 10th Workshop on Logic, Language, Information, and Computation,
Ouro Preto, Minas Gerais, Brazil, 2003, Electron. Notes Theor. Comput. Sci. 84,
Elsevier, 2003, online at http://www.elsevier.nl/locate/entcs/volume84.html.
14
B. KJOS-HANSSEN, A. NIES, AND F. STEPHAN
[3] A. Church, On the concept of a random sequence, Bull. Amer. Math. Soc., 46 (1940),
pp. 130–135.
[4] R. G. Downey, C. G. Jockusch, Jr., and M. Stob, Array nonrecursive sets and
genericity, in Computability, Enumerability, Unsolvability: Directions in Recursion
Theory, S. B. Cooper, T. A. Slaman, and S. S. Wainer, eds., London Math. Soc.
Lecture Note Ser., Cambridge University Press, Cambridge, UK, 1996, pp. 93–104.
[5] S. Ishmukhametov, Weak recursive degrees and a problem of Spector, Recursion
Theory and Complexity (Kazan, 1997), de Gruyter Ser. Log. Appl. 2, de Gruyter,
Berlin, 1999, pp. 81–87.
[6] B. Kjos-Hanssen, W. Merkle, and F. Stephan, Kolmogorov Complexity and the
Recursion Theorem, to appear.
[7] A. Kučera, Measure, Π01 -classes and complete extensions of PA, in Recursion Theory
Week (Oberwolfach, 1984), Lecture Notes in Math. 1141, Springer, Berlin, 1985,
pp. 245–259.
[8] A. Kučera and S. Terwijn, Lowness for the class of random sets, J. Symbolic
Logic, 64 (1999), pp. 1396–1402.
[9] P. Martin-Löf, The definition of random sequences, Inform. and Control, 9 (1966),
pp. 602–619.
[10] D. A. Martin and W. Miller, The degrees of hyperimmune sets, Z. Math. Logik
Grundlag. Math., 14 (1968), pp. 159–166.
[11] A. Nies, Lowness properties and randomness, Adv. Math., 197 (2005), pp. 274–305.
[12] A. Nies, F. Stephan, and S. Terwijn, Randomness, relativization, and Turing
degrees, J. Symbolic Logic, 70 (2005), pp. 515–535.
[13] C.-P. Schnorr, Zufälligkeit und Wahrscheinlichkeit. Eine algorithmische
Begründung der Wahrscheinlichkeitstheorie, Lecture Notes in Math. 218, SpringerVerlag, Berlin, 1971.
[14] S. Terwijn and D. Zambella, Computational randomness and lowness, J. Symbolic
Logic, 66 (2001), pp. 1199–1205.
[15] M. van Lambalgen, The axiomatization of randomness, J. Symbolic Logic, 55
(1990), pp. 1143–1167.
[16] D. Zambella, On Sequences with Simple Initial Segments, ILLC Technical Report
ML 1990-05, University of Amsterdam, Amsterdam, The Netherlands, 1990.